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Strong convergence of infinite color balanced urns under uniform ergodicity
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**Abstract:**
We consider the generalization of the Pólya
urn scheme with possibly infinite many
colors as introduced by Bandyopadhyay and Thacker (2016, 2017).
For countable many colors, we prove almost sure convergence of the urn configuration
under *uniform ergodicity* assumption on the associated Markov chain. The
proof uses a stochastic coupling of the sequence of chosen colors with
a *branching Markov chain*
on a weighted *random recursive tree* as described in
Bandyopadhyay and Thacker (2017).
Using this coupling we estimate the covariance between any two selected colors. In particular, we reprove the limit theorem for the classical
urn models with finitely many colors.